Proof: \(\widehat{\mathbf {X}}_{i}(\mathbf {t})-\mathbf {X}_{i}(\mathbf {t})=\widehat{\mathbf {\mu }}_i-\mathbf {\mu }_i+\widehat{\mathbf {\psi }}_{iM}(\mathbf {t})(\widehat{\mathbf {\rho }}_{iM}-\mathbf {\rho }_{iM})+(\widehat{\mathbf {\psi }}_{iM}(\mathbf {t})-{\mathbf {\psi }}_{iM}(\mathbf {t})){\mathbf {\rho }}_{iM}\) , var\((\widehat{\mathbf {X}}_{i}(\mathbf {t})-\mathbf {X}_{i}(\mathbf {t}))\approx {\mathbf {\psi }}_{iM}(\mathbf {t}) {\rm var}(\widehat{\mathbf {\rho }}_{iM}-\mathbf {\rho }_{iM}) {\mathbf {\psi }}^\top _{iM}(\mathbf {t})\) , due to \(\widehat{\mathbf {\psi }}_{iM}(\mathbf {t})-{\mathbf {\psi }}_{iM}(\mathbf {t})\)  converging to 0 in norm from Propositions 6-7 in [1]} ([1]}). From the distribution of \(\widehat{\mathbf {\rho }}_{iM}-\mathbf {\rho }_{iM}\) , we can infer that \(\widehat{\mathbf {X}}_{i}(\mathbf {t})-\mathbf {X}_{i}(\mathbf {t})\)  follows \(\mathcal {N}_p(\mathbf {0},\mathbf {\psi }_{iM}(\mathbf {t})\mathbf {\Omega }_{M}\mathbf {\psi }^{\top }_{iM}(\mathbf {t}))\)  asymptotically when \(n \rightarrow \infty \) . \(\Box \)
